Solving Inequalities Involving Absolute Values: A Comprehensive Guide

Solving inequalities involving absolute values can be a challenging task, but with a clear understanding of the underlying principles and some strategic approaches, one can effectively determine the solution set. This article will guide you through solving the inequality x - 1(x - 2) ≤ x - 1, exploring various cases and providing a detailed explanation of the solution process.

Introduction to the Problem

Consider the inequality x - 1(x - 2) ≤ x - 1. At first glance, this may seem intimidating, especially with the presence of the (x - 2) term in the numerator. However, by breaking down the problem into manageable cases, we can systematically approach the solution.

Case Analysis

Case 1: x - 1 0

This case occurs when x 1. This is a critical point that needs to be examined separately as it can simplify the original inequality into an equality.

Checking the value of the inequality at x 1:
1 - 1(1 - 2) ≤ 1 - 1
0 ≤ 0

This is a true statement. Hence, x 1 is a solution.

Case 2: x - 1 ≠ 0

In this case, we can divide both sides of the inequality by x - 1 since it is positive. This simplifies the problem significantly. The original inequality becomes:

x - 2 ≤ 1

Further simplifying, we get:

-1 ≤ x - 2 ≤ 1

This can be split into two inequalities:

x - 2 ≥ -1, which simplifies to x ≥ 1 x - 2 ≤ 1, which simplifies to x ≤ 3

Combining these results, we find that:

1 ≤ x ≤ 3

Conclusion and Summary

The solutions from both cases are combined, resulting in the final solution set:

x ∈ [1, 3]

This means that the values of x that satisfy the inequality x - 1(x - 2) ≤ x - 1 are all the real numbers between 1 and 3, inclusive.

Analysis of the Graphical Representation

Graphically, the inequality x - 1(x - 2) ≤ x - 1 can be understood better by considering the parabolic and linear functions involved. The function y x - 1 is a linear function that forms a V-shape with its vertex at (1, 0). Similarly, y x - 2 is also a V-shape with its vertex at (2, 0). These two linear functions intersect with the parabola y x - 1(x - 2). The points of intersection dictate the range of x values that satisfy the inequality.

The shaded area between these functions, including the points themselves, represents the solution set. The points (1, 0) and (3, 0) are the x-intercepts of the parabola, which are also critical in determining the solution range. Therefore, the complete solution set of the inequality is:

x ∈ [1, 3]

This graphical representation provides a visual aid to the algebraic solution, making it easier to understand the nature of the inequality.

Alternative Methods of Solving

Method 1: Absolute Value Approach

Another approach involves the absolute value of the expression. Dividing by |x - 1|, we get:

|x - 2| ≤ 1

This can be split into two inequalities:

x - 2 ≤ 1, which simplifies to x ≤ 3 x - 2 ≥ -1, which simplifies to x ≥ 1

Combining these, we again find:

1 ≤ x ≤ 3

This confirms our previous solution using the initial method.

Method 2: Algebraic Simplification

Starting directly from the inequality and simplifying, we can also avoid the absolute value step. The original inequality simplifies to:

x - 2 ≤ 1

This again results in:

-1 ≤ x - 2 ≤ 1

And combining the results:

1 ≤ x ≤ 3

This reaffirms our solution.

Conclusion

The values of x that satisfy the inequality x - 1(x - 2) ≤ x - 1 are:

x ∈ [1, 3]

This comprehensive approach to solving inequalities involving absolute values can be applied to many similar problems, making it a valuable skill in algebra and mathematical problem-solving.